Question

complete the following. write your answers in simplest form.
(a) volume of the pyramid: (square m^{3})
(b) volume of the prism: (square m^{3})
(c) volume of the pyramid (=square\times) volume of the prism
this equation is true for all triangular prisms and triangular pyramids.
this equation is true only for the triangular prism and triangular pyramid shown above.
this equation is true for all triangular prisms and triangular pyramids with congruent bases and equal heights.

Explanation:

Step1: Calculate the volume of the pyramid

The formula for the volume of a pyramid is \(V=\frac{1}{3}Bh\), where \(B\) is the base area and \(h\) is the height. Given \(B = 16m^{2}\) and \(h=6m\), then \(V=\frac{1}{3}\times16\times6\).

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Step2: Calculate the volume of the prism

The formula for the volume of a prism is \(V = Bh\), where \(B\) is the base area and \(h\) is the height. Given \(B = 16m^{2}\) and \(h = 6m\), then \(V=16\times6=96\).

Step3: Find the relationship between the volumes

Let the volume of the pyramid be \(V_{p}\) and the volume of the prism be \(V_{r}\). We have \(V_{p}=32\) and \(V_{r}=96\). Then \(\frac{V_{p}}{V_{r}}=\frac{32}{96}=\frac{1}{3}\), so \(V_{p}=\frac{1}{3}V_{r}\). This formula \(V_{pyramid}=\frac{1}{3}V_{prism}\) is derived from the general formulas \(V_{pyramid}=\frac{1}{3}Bh\) and \(V_{prism}=Bh\) (when they have the same base area \(B\) and height \(h\)).

Answer:

(a) \(32\) \(m^{3}\)
(b) \(96\) \(m^{3}\)
(c) \(\frac{1}{3}\), and the correct option is "This equation is true for all triangular prisms and triangular pyramids with congruent bases and equal heights."